Investigation of the Factors Affecting the Period of Vibration of a Weighted Cantilever

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  • Topic: Periodic function, Oscillation, Data
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  • Published : December 15, 2012
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Investigation of the factors affecting the period of vibration of a weighted cantilever

Introduction

The aim of the experiment is to find out how the period T of vibration of a weighted cantilever depends on the vibrating length L.

Hypothesis:
Assume the relationship between the vibrating period T and vibrating length L is in accord with the equation: T=kLn, where k and n are both constant values. According to the hypothesis, the vibrating period T and vibrating length L is a linear relationship, so we take logarithms on both sides of the equation. Then, we can get a new equation, which is shown as the following:

logT=logk+nlogL

Variables:
In this experiment, the independent variable is L, the length of vibration. The dependent variable is T, the time taken to complete the oscillation. The controlled Variables is the the amplitude of the oscillation.

Materials required:

1 G-clamps
1 set of standard masses
steel ruler(the cantilever)
stop watch

Methods
Before the experiment, set-up the standard mass at the top of the steel ruler(the cantilever). Step 1: using the G-clamps to set the certain length of vibration. Step 2: pressing the standard mass at the top of the steel ruler and then release it. Step 3: as soon as we release the standard mass, using the stop watch to record the time taken for 10 periods. Step 4: changing the length of the steel ruler and repeat doing the steps above.

Data collection
In this experiment, I choose 30cm, 35cm, 40cm, 45cm, 50cm, and 55cm as the different lengths of the steel ruler. The time taken for completing 10 periods of vibration at each length is shown in the following table.

Table 1- row data
| length(L/m) |time taken(t/s) | | 0.30 |2.1 |1.9 |2.0 |2.0 |2.0 | | 0.35 |2.5 |2.6 |2.6 |2.5 |2.6 | | 0.40 |3.0 |3.1 |3.1 |3.0 |3.1 | | 0.45 |3.6 |3.5 |3.5 |3.5 |3.6 | | 0.50 |4.0 |4.0 |4.1 |4.1 |4.1 | | 0.55 |4.6 |4.7 |4.7 |4.6 |4.6 |

Data processing and Analysis
Table 1 records the time taken for 10 periods, the time taken for 1 period is shown below:

Table 2- time taken for completing 1 period
| length(L/m) |time taken(t/s) | | 0.30 |0.21 |0.19 |0.20 |0.20 |0.20 | | 0.35 |0.25 |0.26 |0.26 |0.25 |0.26 | | 0.40 |0.30 |0.31 |0.31 |0.30 |0.31 | | 0.45 |0.36 |0.35 |0.35 |0.35 |0.36 | | 0.5 |0.40 |0.40 |0.41 |0.41 |0.41 | | 0.55 |0.46 |0.47 |0.47 |0.46 |0.46 |

The uncertainty and the average time taken can be calculated by the function uncertainty of T= (tmax-tmin)/2. The result is shown is the following table.

Table 3- average time taken and uncertainty
|length (L/m) |average time taken (t/s) |uncertainty | | | | | |0.30 |2.00±0.10 |±0.10...
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